On the representation of integers as linear combinations of consecutive values of a polynomial

Let $K$ be a cyclotomic field with ring of integers $\mathcal{O}_{K}$ and let $f$ be a polynomial whose values on $\mathbb{Z}$ belong to $\mathcal{O}_{K}$. If the ideal of $\mathcal{O}_{K}$ generated by the values of $f$ on $\mathbb{Z}$ is $\mathcal{O}_{K}$ itself, then every algebraic integer $N$ of $K$ may be written in the following form:

for some integer $l$, where the $\varepsilon_{k}$'s are roots of unity of $K$. Moreover, there are two effective constants $A$ and $B$ such that the least integer $l$ (for a fixed $N$) is less than $A\,\widetilde{N}+B$, where